Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
949 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(7,13)(11,16)(14,19)(18,20)(21,22,27,31)(23,29,25,28)(24,34,30,26)(32,33,35,36);(6,10)(7,13)(8,15)(18,20)(21,23,27,25)(22,28,31,29)(24,35,30,32)(26,36,34,33);(1,2)(3,4)(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,24,27,30)(22,26,31,34)(23,32,25,35)(28,33,29,36);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,25,27,23)(22,29,31,28)(24,35,30,32)(26,36,34,33);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,26)(22,30)(23,33)(24,31)(25,36)(27,34)(28,35)(29,32)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#45763, #7702426, #7702502, … |
| maximal normal subgroup of | C2×#45763, #7702426, #7702502, … |
| maximal quotient of | #557029, #557036, #557349, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38751);